Asymptotic stability of a nonlinear Korteweg-de Vries equation with critical lengths

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:259

Article

Chu, Jixun^1,2  Coron, Jean-Michel^2,3  Shang, Peipei⁴ 

[1] Univ Sci & Technol, Sch Math & Phys, Dept Appl Math, Beijing 100083, Peoples R China

[2] Univ Paris 06, UMR 7598, Lab Jacques Louis Lions, F-75005 Paris, France

[3] Inst Univ France, F-75005 Paris, France

[4] Tongji Univ, Dept Math, Shanghai 200092, Peoples R China

关键词: Nonlinearity;    Korteweg-de Vries equation;    Stability;    Center manifold;   

DOI  :  10.1016/j.jde.2015.05.010

来源: Elsevier

【 摘 要 】

We study an initial-boundary-value problem of a nonlinear Korteweg-de Vries equation posed on the finite interval (0, 2k pi) where k is a positive integer. The whole system has Dirichlet boundary condition at the left end-point, and both of Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the origin is not asymptotically stable for the linearized system around the origin. We prove that the origin is (locally) asymptotically stable for the nonlinear system if the integer k is such that the kernel of the linear Korteweg-de Vries stationary equation is of dimension I. This is for example the case if k = 1. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free